Surface segregation in binary alloys: A semiempirical approach

Journal of Electron Spectroscopy and Related Phenomena, 37 (1986) 303-318 Elsevier Science Publishers B.V., Amsterdam -Printed in The Netherlands

SURFACE SEGREGATION IN BINARY ALLOYS: A SEMIEMPIRICAL APPROACH

URESH VAHALIA, P.A. DOWBEN and A. MILLER

Department of Physics, Syracuse University, Syracuse, NY 13244 (U.S.A.)

(First received 21 May 1985; in final form 8 September 1985)

ABSTRACT

The preferential segregation concentration of one component in a binary alloy can be estimated, as a function of depth from the surface, from relative X-ray photoemission signal intensities. The details of this semiempirical calculation are outlined. In addition, common approximations applied to these calculations as well as typical experimental errors are analyzed to assess the effects upon the resulting calculated segregation profiles. The semiempirical analysis has been applied to previously published data for Cui7Nss and Fe&rzs binary alloys.

INTRODUCTION

The preferential segregation to the surface of one component of a binary alloy has been the subject of intense study [ 11. Recently, the electron spectroscopy signals and electron mean free paths have been analyzed with the aim of constructing the profile of elemental concentrations from the surface through the selvedge to the bulk [ 2-151. A semiempirical method of determining these concentration profiles has been developed, employing X-ray photoelectron spectroscopy (XPS) intensities for core level emissions at different electron kinetic energies [2] or by comparing relative XPS intensities for different emission angles [ 31.

This method has proven very successful and in good agreement with more direct measurements that combine XPS and Ar+ ion bombardment of the surface [2] . Both methods suggest that the segregation concentration profile as a function of depth from the surface for an Fe-Cr alloy is similar in shape to a Gaussian or exponential curve for equilibrium segregation [2]. The results of these semiempirical calculations have indicated that segregation occurs, not merely for the topmost layer of a surface, but for many atomic layers away from the surface as well [2,3].

Theoretical calculations imply that in most cases, significant segregation

0368-2048/86/$03.60 o 1986 Elsevier Science Publishers B.V.

304

does not occur beyond a few layers below the surface. Kumar [16] has performed calculations for alloys of Ni-Cu and Ag-Au, covering a wide range of values of the bulk concentrations. In all cases, the concentration at the third layer differed from that of the bulk by a few percent or less. (The top layer is defined as the first layer.) Donnelly and King have computed the segregation profile for Cuis Ni,, [17] and for Ags, Au,, and Au4 P& [18]. For Cui, NiaO (loo), (110) and (ill), enrichment of a few percent or less was found at the third layer and insignificant enrichment for deeper layers [17]. The preceding statement also applies to their results [18] for Ag3, Au,, (loo), where they found a deviation from bulk values at the fifth layer of about one percent. The enrichment of Au in Au4 Pt% (100) was about one percent at the fifth layer.

In the semiempirical calculations based on the XPS data, approximations were made, the validity of which must be scrutinized. Moreover, it is important to analyze the effect of typical experimental errors on the resulting calculated segregation profiles. In this communication, we will outline the method for calculating the segregation concentration profile (as a function of distance from the surface) from experimental XPS data. We will also show the effect of typical experimental errors and approximations on the calculated profile parameters, thus providing experimentalists with a framework to assess the reliability of their data. We will apply the analysis to previously published results on Cu,, Niss (loo), Cui, Nis, (111) [31 and Fe72 Crzs (110) [2] that were annealed at 800 K [5,6] and at 750 K [2], respectively, to achieve equilibrium segregation, and then quenched to room temperature.

CALCULATION OF PROFILE PARAMETERS

Consider a miscible substitutional binary alloy without precipitation of either element. We may distinguish between two categories of such alloys: clustered and ordered.

For the clustered alloy, the number of like-atom nearest neighbor bonds exceeds the number which would occur if the atoms randomly filled the lattice (chemical composition fixed). Clustering will occur if the magnitude of the energy of attraction between neighbors of unlike atoms is less than the size of the arithmetic average (taken over the two constituents) of the like-atom nearest neighbor energies. (The “ideal solution” corresponds to the case in which these two characteristic energies are equal.) It is then reasonable to expect that the segregation profile is monotonic. Evidence that Ni-Cu alloys are (weakly) clustering can be found in Kumar [16] and in Donnelly and King [ 171.

If, however, the magnitude of the energy of attraction between nearest neighbors of unlike atoms is greater than the size of the arithmetic average of

305

the like-atom energies, then the alloy is ordered, and the segregation profile may be expected to be oscillatory. An example is the evidence cited in refs. 16 and 18 that Au-Ag alloys are ordered.

In this paper, the discussion will be restricted to clustered alloys. Dowben et al. [2] performed argon ion bombardment experiments on an FeT2 Cr2s alloy and obtained a segregation profile, without any presupposition as to its analytical form. The results indicate a monotonically varying profile, describable by various functional forms. In this work, we choose exponential and Gaussian forms for the profiles, both for the Fe,* Crzs data of ref. 2 and for the Cui, Niss results of refs. 5 and 6. These two forms were used earlier by Dowben et al. [2] to analyze their XPS measurements on Fe,2 CrZs; the resulting profiles were found to be in good agreement with the profile obtained from the argon ion bombardment data.

If an exponential or Gaussian shape is assumed, then two characteristic parameters must be obtained from XPS data. These parameters can be labeled 6 (the segregation at the topmost layer) and G (the segregation depth, expressed in units of the distance d between layers).

Let A and B denote the two constituent elements of the binary alloy, with A chosen as the element that segregates to the surface. (Thus, A is copper for copper-nickel alloys and is chromium for iron-chromium alloys.) The fraction fi of element A for the ith layer is given by

fi = b + 6e-i/G (la)

for the exponential profile, where b is the bulk fraction for element A. From knowledge of the profile form fi, one can calculate C, the apparent surface concentration (or relative intensity) of element A for a particular core level and for a fixed electron emission angle. C is defined as the normalized electron emission intensity for element A divided by the sum of the normalized electron emission intensities for both components. The normalized intensity is the actual emission intensity divided by the corresponding differential cross section for emission. (The actual emission intensities include a correction for changes in the analyzer efficiency with energy.)

The expression determining C is then

c= izo fie-“aA

izo fie- i/AA + igo (1 - fi)emilhB

(2)

Here, AA and ha are the effective mean free paths (in units of d) for photo- electrons emitted from the particular core level, for components A and B, respectively. Note that the relation between the effective mean free path X and the actual mean free path X,, is X = X0 cos 0, where 8 is the emission

306

angle, relative to the surface normal (with the subscripts A or B on the mean free paths suppressed). For the iron-chromium data [2] , the emission is in the normal direction so that X = X0. Also, note that the use of normalized intensities serves to eliminate the differential cross sections from the equations.

In studying the Cui, Niss alloy, the XPS 2p3,1 signals were obtained for two different emission angles (45 and 75”) [ 5,6]. Resulting are two pairs of values for XcU and XNi, one for the CU 2~3,~ level and the other pair for the Ni 2~s,~ level. Hence, eqn. (2) generates two equations, corresponding to the two measured values of C.

Previous semiempirical calculations [2, 31 have made an approximation based on the fact that the kinetic energies of the electrons for one constituent from a given core level are close to those of the same core level for the other constituent. Thus, each core level can be described by a single ‘average’ effective mean free path X = hA = ha. This approximation allows the following simplification of eqn. (2) for the case of the exponential profile

c= b+ 1 - e-l/h

1 - e(-l/h-l/G) (34

For the experiments of the Cui, Nis3 alloy [ 5, 61, the average effective mean free path X differs for the two emission angles, yielding two separate equations which can be solved simultaneously for 6 and G.

The procedure is similar for a Gaussian profile. The form

fi = b + 6e-izlG2 (lb)

replaces eqn. (la). From eqn. (2), the apparent surface concentration is

G = b + a(1 -e-l/h) 2 e-i*lG*e-l/a

i=O (3b)

if the single mean free path approximation is used. Employing eqns. (3a) and (3b), a computation for 6 and G has been

performed for the Cul, Niss (111) and Cui, Niss (100) data [5, 61. (For the case of the exponential profile, the results for 6 and G are more easily obtained from eqn. (Al) of the Appendix. The latter equation is obtained by solving simultaneously the pair of equations obtained from eqn. (3a) to get explicit results for 6 and G.) The results are listed in Table 1.

For the FeTz Cr,, (110) data [2], a similar calculation yields the results displayed in Table 2. In this case, each core level (2p or 3p) yields one of the two equations whose simultaneous solution determines 6 and G.

ERROR ANALYSIS

An effective way of determining the effect of experimental error is via contour plots. The effect of errors in the empirical results for C on the

307

TABLE 1

PROFILE PARAMETERS FOR Cu17 Niss (SINGLE MEAN FREE PATH APPROXI- MATION)

Parameter Cu17 Niss (111) Cui7 Nisa (100)

b = bulk fraction of Cu x = average mean free path in atomic layers

45’ emission 7 5’ emission

C = fractional Cu signal 45’ emission 75’ emission

Exponential profile parameters Segregation depth G (atomic layers) Top layer segregation 6

Gaussian profile parameters G (atomic layers) 6

0.17

2.7 0.99

0.45 0.52 0.69 0.71

0.756 2.272 0.739 0.675

0.984 2.501 0.724 0.630

0.17

3.1 1.1

TABLE 2

PROFILE PARAMETERS FOR FeT2 Crza (SINGLE MEAN FREE PATH APPROXI- MATION)

Parameter 2p case 3p case

0.28 0.28 5.4 8.0

C 0.63 0.58 Exponential profile: G = 10.60, 6 = 0.504 Gaussian profile: G = 11.846,6 = 0.436

calculated values of G and 6 can be expressed by meter

defining the error para-

E = r&pt. - C(G, S)] * + [C&. - C’(G, S)] * (4)

In eqn. (4), C,. and c&t. are the empirical values of the apparent surface concentration for the two measurements, each corresponding to different effective mean free paths. Similarly, C(G, 6) and C’(G, 6) are the theoretical profiles from eqn. (2), generated from the effective mean free path corresponding to the measured value.

For the Cu,, Niss data [5, 61, contours of constant G have been plotted in the G-8 plane for E values of 1, 2, and 3% error (Fig. 1). The contours give a good picture of the range of error that may be expected in G and 6 for

308

Error Contours for Cu,,Nie3

I .oo CuNi (Ill) CuNi (100) 1 I 8 1 l.OO- I I 1 I A C

0.90 - Exp - 0.90- EXP -

- 0.80-

- 070-

0.60 - - 0.60-

0.50 I , 1 I ’ 0.50. I I I 0.0 0.40 0.80 1.20 1.60 2.00 0.0 I.0 2.0 3.0 4.0

D Gsn

i

0.50- 0.501 0.0 0.50 IO0 1.50 2.00 0.0 I .o 2.0 3.0 4.0

G (atomic layers) G (atomic layers)

Fig. 1. Error contours for Cu17 NisJ (100) and (111) for exponential and Gaussian profiles. Contours are plotted in the G-6 plane for 1, 2 and 3% error in the data.

a known inaccuracy in the experimental data. The plots, however, tend to overestimate the error. This is because in typical experimental conditions, the error is not of a completely random nature, and certain types of errors are much more likely. We therefore prefer to employ a more effective method, plotting curves of constant C in the G-6 plane.

For a fixed emission angle, eqn. (3) gives a curve in the G-6 plane, corresponding to the pairs G, 6 yielding the observed relative intensity C. Each of the graphs in Fig. 2 contains two families of three approximately parallel curves. For each family of three curves, the curve with intermediate value of 6 corresponds to the actual reported data. The other two curves in the family correspond to an assumed plus 2% deviation in C (curve with largest 6) and an assumed minus 2% deviation in C (curve with smallest 6). This enables us to see how an error in the experimental data for C affects the calculated values of G and 6.

Figure 2 shows these ‘G-DELTA’ plots for the (111) and (100) surfaces of Cu17 Niss, for exponential and Gaussian profiles. Five of the intersection points are labeled. The point x corresponds to the best fit solution, whereas

309

G -DELTA plots for Cu,,Nie3

l.OO( CuNi (Ill)

I I I , I.001 CUNI (100)

\\\’ I I 1

0.90 0.90

Mz 0.80

0.70

0. 70 0.60

0. 60 0.50 0.0 0.5 I.0 1.5 2.0 0.0 I.0 2.0 3.0 4.0

1.00 1.00

0.90 0.90

is 0.80

;-Q 0.80

ii 0.70 n

0.70 0.60

0.60 0.50 0.0 0.5 1.0 I .5 2.0 0.0 I .o 2.0 3.0 4.0


Fig. 2. G-DELTA plots for 0.1~~ Niss (100) and (111). For each value of C there are three approximately parallel curves. The curve with intermediate value of 6 corresponds to the actual data. The curves with largest and smallest 6 correspond to an assumed plus 2% and minus 2% deviation in C, respectively. The intersections of these lines are marked by the letters a, b, c, d and x, with the point x denoting the solutions for the actual data.

points a, b, c, and d correspond to solutions for a 2% error in both data points. Figure 3 shows the segregation profiles corresponding to the solutions of G and 6 pairs identified in Fig. 2 as a, b, c, d, and x.

While, at first glance, it appears that the profile parameters are extremely sensitive to errors in measurement, we must examine the likely types of errors. In most typical cases, the errors in measurement would be such that the error would have the same sign for both the emission angles. This would be the case if, for example, a constant background signal was superimposed on both the Cu and Ni.

Now an error in the same direction for the two emission angles corresponds to points b or d in Fig. 2, or in general, somewhere along the line b-x-d. Along this line it is clear that the change in G or 6 is very small, and hence we can conclude that typical experimental errors have little effect on

310

Segregation Profiles for CulTNies

100, CuNi (Ill)

I I I , 1001 CuNi (100)

1 1 I I

60

80

5 z 60

E 8 40

$

- 20

0 0.0 1.0 2.0 3.0 4.0 0.0 2.0 4.0 6.0 8.0

Depth in atomic layers Depth in atomic layers

Fig. 3. Segregation profiles for Cul, Nisa (100) and (111). The profiles are plotted for the five points a, b, c, d, x marked in the G-DELTA plots.

calculated results. Errors which have different signs for the two emission angles (such as point a or c in Fig. 2) are both atypical and unlikely. If the experiments are done carefully, they would be very small. This is consistent with the data on Fe,2 Crzs in ref. 2, in which the apparent surface concentrations determined by XPS varied somewhat from experiment to experiment, but the difference in concentration determined from the 2p and 3p core levels was invariably 5%.

THE SINGLE MEAN FREE PATH APPROXIMATION

In the preceding analysis for Cui, NisJ, we made the approximation that the mean free paths for the copper and nickel electrons were approximately equal, and could be replaced by a single effective average mean free path A. This value of h, of course, is different for the two emission angles. This

311

TABLE 3

EXACT ANALYSIS OF PROFILE PARAMETERS

Cu1, Niss (111) 45O

k!U 2.58 hNi 2.34 x *v 2.71

Result

G (exact) G (approx.) 6 (exact) 6 (approx.)

Cul, Niss (100)

Result

hCU 2.94 1.08 xNi 3.23 1.18 h a” 3.09 1.13


Fe72 Cr28

hFC3

hr h a”

Result


Exponential

0.930 0.756 0.739 0.739

45O

Exponential

2.592 2.272 0.680 0.675

2P

5.07 5.70 5.39

Exponential

24.910 10.598

0.387 0.504

75O

0.945 1.04 0.995

Gaussian

1.156 0.984 0.715 0.724

75O

Gaussian

2.790 2.501 0.634 0.630

3P

8.04 8.11 8.08

Gaussian

21.277 11.846

0.356 0.436

approximation is not strictly necessary, and we can as well use the exact eqn. (2) instead of the approximate eqn. (3). The result is only a little extra mathematical complexity. The needed generalization of eqn. (3a) is listed as eqn. (A2) in the Appendix and gives the simultaneous equations determining G and 6, for the exponential profile. For Cu17 Niss, however, the exact analysis yields results that are very close to the approximate analysis, as shown in Table 3, and plotted in Fig. 4. Thus, it seems desirable to use the approximation for the copper--nickel data, as was done in an earlier study 131.

The case of Fe,? Crzs requires separate discussion. Table 3 also includes a comparison of the values of the profile parameters for the exact and

312

Effect of Single Mean Free Path Approximation on CuNi Profiles

100, CuNi (I I I) CuNi (100)

I 1 I , 10011

3 ‘i; E

E n

0 I I I 0 1 I I 0 I .o 2.0 3.0 4.0 0 2.0 4.0 6.0 8.0

100 I I I IOO- I 1 I

B D

- 20-

0- o- 0 I. 2.0 3.0 4.0 0 2.0 4.0 6.0 8.0

Depth in atomic layers Depth in atomic layers

Fig. 4. Effect of single mean free path approximation on Cu17 Niss surface segregation profiles. In each graph, curve 1 represents the profile for the exact solution, while curve 2 represents the solution for the single mean free path approximation.

approximate methods for Fe,2 Cr2s. In Fig. 5, the profiles for the single mean free path approximation are compared with the exact results. It can be seen from both Table 3 and Fig. 5 that the approximation gives results that differ markedly from the exact solution. What then, is the difference in the two cases (Fe,2 Cr2s as compared to Cul, Nis3)? It is not that the mean free paths for Fe and Cr are very different. In fact, the relative difference between the two mean free paths is about the same for the two alloys.

The answer lies in two separate considerations. First, when G is large, the value of G is very sensitive to changes in the value of exp (- l/G). Hence, any errors in the value of exp (- l/G) due to the approximation result in a relatively large error for gentle profiles such as that for FeT2 Cr2s, as compared to relatively steep profiles, like that of Cul, Nies.

Secondly, when the two readings correspond to differing emission angles, (same core level), then the ratios AA/AA’ and Xs/XBl are both equal to cos~/cosO ‘. (0 and 0 ’ are the emission angles for the two readings.) This

313

Effect of Single Mean Free Path Approximation on Fe&-,, Profiles

I I I Gsn

go-

t

I I I

Exp

go-

IO. I I I 0 IO 20 30 40

Depth in atomic layers

Fig. 5. Effect of single mean free path approximation on Fen Crzs (110) surface segregation profiles. In each graph, curve 1 represents the profile for the exact solution, while curve 2 represents the solution for the single mean free path approximation.

reduces the error caused by the single mean free path approximation. The small deviation between exact and approximate profiles is displayed in Fig. 4. For the case of Fe,2 Crzs (use of differing core levels), the ratios differ for the two elements. There is no cancellation in the error caused by equating hA to hA 1 and XB to haa and the approximation is not reliable. In general, the approximation must be used with caution, and its applicability for one system does not mean it is useful for a different one.

In Fig. 6, the G-delta curves and segregation profiles for FeT2Crzs are displayed. As is the case for Cui7 Niss , there are relatively small deviations in G and 6, due to the most likely types of error.

DISCUSSION

As is seen from Tables l-3, segregation is not restricted to the surface but extends considerably into the selvedge region. For the case of Fe,2 Crzs, the segregation depth is as much as 25 atomic layers, implying significant

314

segregation up to 40 atomic layers. This figure is considerably higher than the results published previously [2] , which showed segregation to about 20 atomic layers. The earlier calculations used the single mean free path approximation, which is not valid for this case.

The contour plots and the G-DELTA plots together provide an adequate framework to analyze the reliability of the data. They indicate that typical experimental errors do not affect the results significantly, and that the error in the results can be gauged fairly accurately. The methods outlined so far, however, do not provide any criteria to select one particular profile shape over any other. We have no way of deciding whether a Gaussian, exponential, or some totally different profile, best fits the data.

The reason is that when the experiment is such as to provide just two values of C, then eqn. (2) or (3) can be solved exactly to give values of G and 6 which fit the data perfectly. In other words, there is always one point on the contour plots corresponding to E = 0. This suggests that XPS experiments should be performed which yield more than two data points, each corresponding to a different mean free path. For example, data could be obtained for three different emission angles, or for two different emission angles with two core levels of emission. Then the contour plots will have a minimum value of e, which we can call emin. Different choices of the profile shape will have different values of E,~, and the one with the lowest value is the most desirable profile for that alloy.

The analysis here has focused on the XPS studies on Fe,* Crzs of Dowben et al. [2] and on Niss Cu i7, performed by Brundle and Wandelt [5]. We note briefly other studies obtaining composition profiles in alloys. Most of these profiles are based upon measurement of relative intensities of electrons emitted with differing mean free paths.

A study by Webber et al. [4] on Cu, Nig, using Auger electron spectroscopy (AES) found significant enrichment of Cu in the top layer, with the composition dropping to nearly bulk levels in the next two layers. For deeper layers, there was evidence of a rise in Cu composition for the next three or four layers, before the bulk level was regained. A UV photoemission study by Ling et al. [ 71 on Ni9s Cu,, (110) was performed for two surfaces annealed at different temperatures. In both cases evidence was found for an oscillation in the composition profile. Simple models indicated appreciable segregation in the first five layers.

Kuijers and Ponec [8] used AES on Ni-Cu alloys; the top layer concentration was determined independently from low energy ion scattering, while the second layer concentration was determined from the AES data.

Watanabe et al. [9] assumed an exponential profile in their AES study of Cu-Ni alloys; significant segregation in the first four layers was found. In the AES experiments of McDavid and Fain [lo] on Cu-Ni alloys, results for the concentration were obtained for the first and second layers. The calculation assumed that the deeper layers have attained the bulk

315

concentrations. Segregation in the selvedge has not been uniformly observed however, for copper-nickel alloys. In an atom-probe field ion microscopy study of an Niss Cus alloy, Ng et al. [ll] found substantial enrichment of Cu only in the first layer.

In studies of other alloys, King and Donnelly [12] performed an AES study on Ag-Au alloys containing 10, 20 and 30% silver. Results for the concentration of the first and second layers were found for surfaces (loo), (110) and (111) by minimizing the sum of the square deviations of observed and calculated Auger peak intensity ratios for 14 Auger transitions. It was assumed that bulk values for the concentration applied beyond the second layer. Sedlacek et al. [13] examined Pt-Ni alloys with AES and found platinum enrichment at the surface; the concentrations were determined for the first ten layers.

Iron alloy studies include a determination of the concentration profile of a FeT4 Crz6 polycrystalline alloy by Pons et al. [14], using AES and Ar+ ion sputtering. Enrichment of Cr in the first 4 or 5 layers was found. Dowben and Grunze [15] obtained profiles of the segregation of carbon to the Fe (100) surface, using a comparison between the carbon KLL AES signal and an XPS C 1s core level signal.

SUMMARY

The profile parameters 6 and G are defined in eqns. (la) and (lb), for the exponential and Gaussian forms, respectively. The parameter 6 is the fractional enrichment of the segregating element at the topmost layer, while G is the distance (in units of the spacing, d, between layers) for which the enrichment falls to l/e of its value at the top layer. Equation (2) gives the apparent surface concentration C of the segregating component for a given pair of effective mean free paths XA and hB .

If data a-.e available for more than one pair of effective mean free paths hA and Xs, then a profile for penetration of segregation into the selvedge can be deduced. This can be obtained by varying the electron emission angle or by examining emission from different electron core levels. In particular, when two values for the apparent surface concentration are known, the profile forms of eqns. (la) and (lb) are exactly determined, as each has two parameters.

Equations (3a) and (3b) simplify eqn. (2) for the cases of the exponential and Gaussian profile respectively. Equation (3), however, is valid only if the single mean free path approximation XA = XB is a good approximation. Otherwise, it is necessary to insert eqn. (1) into eqn. (2) and then solve the resulting two equations (generated from the two observed values of C) simultaneously by numerical means. For the case of the exponential profile, the series in eqn. (2) can be summed by analytical means and the two

316

G-DELTA curves for Fe72Cr2e


Segregation Profiles for Fe,,C&e

0 IO 20 30 40 50 60


70 C

50 X

k

a 30

IO1 ’ ’ ’ ’ ’ ’ 0 IO 20 30 40 50 60


Fig. 6. G-DELTA plots and segregation profiles for FeT2 Crzs (110). The exact analysis has been used here, in contrast to the use of the single mean free path approximation in Figs. l-3 for Cu l7 Niss .

simultaneous equations simplify to the form given in eqn. (A2) of the Appendix. In particular, when the single mean free path approximation is valid, eqn. (A2) or eqn. (3) can be solved to yield explicit values for 6 and G. The results are listed in eqn. (Al) of the Appendix.

Numerical results for 6 and G are displayed in Tables l-3, for the data of refs. 2, 5 and 6. In all cases, the results for G indicate substantial penetration of the segregation into the selvedge region, and thereby support similar conclusions of earlier work [2,3] .

There are cases in which the single effective mean free path approximation is shown to be quite good (see Fig. 4). However, we have also shown that in some examples the approximation significantly underestimates the result for G and does not provide an accurate method for calculating the segregation profile (see Fig. 5). The approximation may give poor results when the profile is very gentle and/or when the ratios of the effective mean free path

317

for the two readings depend markedly on the constituent element, as detailed earlier.

Error contours for the Cui, NisS data [5, 61 given in Fig. 1 indicate the directions in the S, G plane for smallest and greatest changes in the results for 6 and G, corresponding to small variations in the experimental values of the apparent surface concentrations. In the G-Delta plots for these samples (Fig. 2) the change in the values of G and 6 is given for * 2% changes in each of the two values of C. These results indicate relatively small deviations in the numerical results for G and 6 due to the most likely types of error (both values of C overestimated or both underestimated). This fact can also be seen from inspection of the profiles themselves (Fig. 3). For the Fe,2 Cr2s sample, Fig. 6 displays the G-DELTA curves and profiles for the measured values of C, as well as for f 2% error in each value. The results are consistent with the remarks just made for Cui7 Nisj as to the nature of the errors in 6 and G from the likely errors in C.

In the third paragraph of the Discussion, it was pointed out that XPS experiments are now feasible whose outcome would indicate a preference for the form of the profile (exponential, Gaussian, or other form). In addition, theoretical studies from a microscopic viewpoint are needed, to explain the extensive penetration of segregation into the selvedge region.

APPENDIX

In this Appendix, we list relations determining the profile parameters 6 and G that are generally simpler to use than eqns. (2) and (3), for the case of the exponential profile.

For the cases where the single mean free path approximation is justified, eqns. (3a) or (3b) each generate two equations to be solved for 6 and G, appropriate for the exponential or Gaussian profiles, respectively. For the exponential form, the two equations obtained from eqn. (3a) can be solved, yielding explicit results for 6 and G.

The results can be displayed by letting C and A in eqn. (3a) correspond to a measurement for one value of effective mean free path, while C’ and A’ denote the values for the other effective mean free path. Solving the two equations yields the results for G and S listed below

e-UG = xD' -x'D + D -D'

xD - x'D' + xx'(D) - D) (Ala)

6= DD’(x -x’)

xD - x’D’ + xx’(D’ -D) @lb)

In eqn. (Al), x = exp (- l/X) and x’ = exp (- l/h’). Also, D = b -C and D’ = b - C’ are the empirical values for the enrichment.

318

For the cases in which the single mean free path approximation is not warranted, eqns. (la) or (lb) must be inserted into eqn. (2), keeping XA # ha. For the exponential profile, the result is

c = ‘[l + (X/Y)] -l (A2a)

X = YAZA I(1 -brZB -YB~I Wb)

Y = YB%(~~A + YAP) WJc)

In eqns. (A2b) and (ARC), yA and zA are defined by yA = l- exp (- l/XA ), zA = 1 - exp (- G-r - ?I>’ ); the definitions of ya and zs are analogous, with A replaced by B. Equation (A2a) is then used twice, for the two measured values of C, corresponding to the two pairs of values for hA and ha; the resulting pair of equations determines 6 and G.

On the other hand, for the Gaussian form, the sums on i cannot be obtained in closed form. Equation (lb) must be inserted into eqn. (2), and the resulting two equations solved simultaneously by numerical means.

REFERENCES

1 F.F. Abraham and C.R. Brundle, J. Vat. Sci. Technol., 18 (1981) 506. 2 P.A. Dowben, M. Grunze and D. Wright, Surf. Sci., 134 (1983) L524. 3 P.A. Dowben, Phys. Rev. B, 30 (1984) 7278. 4 P.R. Webber, C.E. Rojas, P.J. Dobson and D. Chadwick, Surf. Sci., 105 (1981) 20. 5 C.R. Brundle and K. Wandelt, J. Vat. Sci. Technol., 18 (1981) 537. 6 K. Wandelt and C.R. Brundle, Phys. Rev. Lett., 46 (1981) 1529. 7 D.T. Ling, J.N. Miller, I. Lindau, W.E. Spicer and P.M. Stefan, Surf. Sci., 74 (1978)

612. 8 F.J. Kuijers and V. Ponec, Surf. Sci., 68 (1977) 294. 9 K. Watanabe, M. Hashiba and T. Yamashina, Surf. Sci., 61 (1976) 483.

10 J.M. McDavid and S.C. Fain, Jr., Surf. Sci., 52 (1975) 161. 11 Y.S. Ng, T.T. Tsong and S.B. McLane, Jr., Surf. Sci., 84 (1979) 31. 12 T.S. King and R.G. Donnelly, Surf. Sci., 151 (1985) 374. 13 J. Sedlacek, L. Hilaire, P. Legare and G. Maire, Appl. Surf. Sci., 10 (1982) 75. 14 F. Pons, J. Le Hericy and J.P. Langeron, J. Microsc. Spectrosc. Electron., 2 (1977)

49. 15 P.A. Dowben and M. Grunze, J. Electron Spectrosc. Relat. Phenom., 28 (1983) 249. 16 V. Kumar, Phys. Rev. B, 23 (1981) 3756. 17 R.G. Donnelly and T.S. King, Surf. Sci., 74 (1978) 89. 18 T.S. King and R.G. Donnelly, Surf. Sci., 141 (1984) 417.

Documents

Surface segregation in binary alloys: A semiempirical approach